Which Sets below may be considered as Universal Set(s) for the previous Sets?

Universal Set

If there are some sets under consideration,then there happens to be a set which is a superset of each one of the given sets. Such a set is known as the universal set and it is denoted by 𝕌.
e.g.,
(i) Let A={2,4,6}, B={1,3,5} and C={0,7} Then, 𝕌={0,1,2,3,4,5,6,7} is an universal set.
(ii) For the set of all integers, the universal set can be the set of rational numbers or the set of real numbers.

[Definition] In many discussions all the sets are considered to be subsets of one particular set. This set is called the universal set for that discussion.

The Universal set is often designated by the script letter 𝕌 (or by 𝕏).
Universal set in not unique,and it may change from one discussion to another.

Example: If P={0,2,7}, Q={3,5,6}, R={1,8,9,10} then the universal set can be taken as the set.

𝕌={0,1,2,3,4,5,6,7,8,9,10}.

⛲ Question 1. Given the sets S={1,3,5},T={2,4,6} and V={0,2,4,6,8}. Which of the following may be considered as universal set(s) for all three sets S, T, and V?
(i) Ø
(ii) {0,1,2,3,4,5,6,7,8,9,10}
✍ Solution:
We know that, universal set for sets S, T, and V is superset of S, T, and V i.e.,universal set contains all elements of S, T, and V.
(i) Ø cannot be considered as universal set.
(ii) {0,1,2,3,4,5,6,7,8,9,10} is the universal set for the given sets S, T, and V as all the elements of sets S, T, and V are in this set.

⛲ Example 1. Given the sets W={1,3,5}, X={2,4,6} and Y={0,2,4,6,8}, which of the following may be considered as universal set(s) for all the three sets W, X, and Y
(i) {0,1,2,3,4,5,6}
(ii) Ø
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
✍ Solution:
(i) It can be seen that W⊂{0,1,2,3,4,5,6}
X⊂{0,1,2,3,4,5,6}
However,Y⊄{0,1,2,3,4,5,6}
Therefore,the set {0,1,2,3,4,5,6} cannot be the universal set for the sets W, X, and Y.
(ii) W⊄Ø, X⊄Ø, Y⊄Ø
Therefore,Ø cannot be the universal set for the sets W, X, and Y.
(iii) W⊂{0,1,2,3,4,5,6,7,8,9,10}
X⊂{0,1,2,3,4,5,6,7,8,9,10}
Y⊂{0,1,2,3,4,S,6,7,8,9,10}
Therefore,the set {0,1,2,3,4,5,6,7,8,9,10} is the universal set for the sets W, X, and Y.
(iv) W⊂{1,2,3,4,5,6,7,8}
X⊂{1,2,3,4,5,6,7,8}
However, Y⊄{1,2,3,4,5,6,7,8}
Therefore, the set {1,2,3,4,5,6,7,8} cannot be the universal set for the sets W, X, and Y.

Universal Set
Another important set is a universal set.

Definition: Universal Set
A universal set, symbolized by 𝕌, is a set that contains all the elements for any specific discussion.

When a universal set is given. only the elements in the universal set may be considered when working the problem. If,for example,the universal set for a particular problem is defined as 𝕌={1,2,3,4,…,10}. then only the natural numbers 1 through 10 may be used in that problem.

⛲ Ex2. Given the universal set={x:x∈ℕ and x<20},find: Z={x:x=3p; p∈ℕ}
✍ Solution:
Universal set 𝕌={x:x∈ℕ and x<20}

={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19}
Z={x:x=3p; p∈ℕ}
x=3p
If p=1,then x=3⋅1=3
If p=2,then x=3⋅2=6
If p=3,then x=3⋅3=9
If p=4,then x=3⋅4=12
If p=5,then x=3⋅5=15
If p=6,then x=3⋅6=18
Z={3,6,9,12,15,18}

🌈 Universal set and Complements

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